All Questions

7
votes
1answer
115 views

Rothberger property for finite covers

Let us recall that a topological space $X$ has the Rothberger property if for any sequence $(\mathcal U_n)_{n\in\omega}$ of open covers of $X$ there exists a sequence $(U_n)_{n\in\omega}\in\prod_{n\...
2
votes
2answers
88 views

Examples of group $G=N \rtimes H$ where $N$ and $H$ are as below

I am searching for examples of connected locally compact group $G = N \rtimes H$, where $N$ is a simply connected nilpotent non-abelian Lie group, $H$ is linear reductive and $H$ operates on $N$ ...
2
votes
1answer
21 views

Reference request: existence of a subgroup of $G(\mathcal O_k)$ that is “uniform” across $P \overline{N}$

Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $P_0$ be a minimal parabolic subgroup of $G$ containing a maximal split torus $A_0$. Let $K$ be a maximal compact open subgroup ...
6
votes
1answer
63 views

How to compute the Kahler potential of a Sasaki metric

The Question Given Hessian manifold $M$, there is a natural Kahler structures on $TM$. Is it possible to write the Kahler potential of these in terms of the Hessian potential? Background To ...
3
votes
3answers
233 views

Intersection of $\{2^a 3^b 5^c 7^d\}$ and its translates

Let $S$ be the set of positive integers of the form $2^a3^b 5^c 7^d$. I need information about the cardinality of the intersection of $S$ and its translates. In particular, is $S \cap (S+t)$ ...
5
votes
1answer
269 views

What is $\mathrm{O}_q/\mathrm{SO}_q$ if $q$ is a quadratic $\mathbb{Z}$-form which is degenerate?

Any binary quadratic $\mathbb{Z}$-form $q$ induces a symmetric bilinear form $$ B_q(u,v) = q(u+v) - q(u) -q(v) \ \ \forall u,v \ \in\mathbb{Z}^2 $$ and it is considered non-degenerate (over $\mathbb{...
1
vote
0answers
47 views

Given a set of vertices of the Boolean cube, what is the equivalent union with least number of faces?

For example. Suppose you are compacting binary using the symbol X. Like a pattern that expands to (0,1). Example. BINARY SET 0000 0010 1000 1010 MINIMAL ...
8
votes
3answers
427 views

Stiefel-Whitney total class with prescribed zeros

First things first, I am aware of the existence of this topic. It's related, but old and my question hasn't been discussed there. So I hope it's not wrong to start a new topic. I'm currently ...
0
votes
0answers
12 views

Name for mappings that are “not quite projections”

Is there a known name for the following definition? Consider topological spaces $X$, $Y$ and $f: X \rightarrow Y$ a continuous mapping. Then, $f$ is an "almost projection" if there is a topological ...
8
votes
1answer
168 views

Functoriality of (co)limits in $\infty$-categories

I have some questions about the functoriality of (co)limits in $\infty$-categories, say in the framework of Lurie's Higher Topos Theory. From the general stuff about Kan-extensions (HTT 4.3.2.6) ...
10
votes
3answers
225 views

Minimizing geodesics in incomplete Riemannian manifolds

Let $(M, g)$ be a Riemannian manifold, not necessarily complete. Let $x$ be a point in $M$, and let $r>0$ be such that the exponential map $\operatorname{exp}_x$ is defined on an open ball $B=B(0,r)...
3
votes
1answer
162 views

Foliation with a compact leaf

Let $M$ be a closed oriented manifold, and $F$ be a fixed foliation of $M$. We assume the dimension and codimension of $F$ are both greater than $1$. Q Under what condition, we can say that $F$ ...
2
votes
1answer
43 views

Spectral density of $D + XX^T$

Let $D$ be a fixed diagonal matrix with real entries, and $X$ a random $m\times n$ matrix. More precisely, the entries $X_{ij}$ are real independent and identically distributed. It can be shown that ...
1
vote
0answers
38 views

Uniqueness of limits and compactness implies closure

It is not difficult to prove that in a Hausdorff topological space every compact set is closed, and almost trivial that if in a topological space X every compact set is closed then X is T1. As ...
-1
votes
0answers
68 views

Specific examples of an algebraic closure of a finite field [on hold]

I'm struggling to understand the concept of algebraic closure for finite fields. Are there specific examples I can use to get an intuitive understanding? What sorts of elements do the algebraic ...
4
votes
1answer
79 views

Smooth structure on the space of sections of a fiber bundle and gauge group

Let $\xi$ be a fiber bundle $F\hookrightarrow E\to B$ (where every space is smooth, T2 and second countable), let $\Gamma(\xi)$ be the space of smooth sections. We can complete $\Gamma(\xi)$ with ...
1
vote
2answers
88 views

Given an integer lattice, how to count the number of points whose norm is smaller than some bound $M$?

Let $\mathbf{b}_1, \mathbf{b}_2, ..., \mathbf{b}_n$ be linearly independent $m$-dimensional vectors whose entries belong to $[0, M] \cap \mathbb{Z}$, for some $M \in \mathbb{N}^*$. Of course, $n \le ...
3
votes
0answers
45 views

Theory of surfaces in $\mathbb{R}^3$ as level sets

Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I ...
1
vote
0answers
36 views

Matching Stochastic Flows

Let $\nu = (\nu_t)_{t \in [0,T]} \in C( [0,T], \mathcal{P}_2(\mathbb{R}) ) ,$ where $\mathcal{P}_2(\mathbb{R})$ denotes the space of probability measures with finite moment equipped with the ...
6
votes
1answer
130 views

Tiling with incommensurate triangles

Say that two triangles are incommensurate if they do not share an edge length or a vertex angle, and their areas differ. Suppose you'd like to tile the plane with pairwise incommensurate triangles. I ...
1
vote
0answers
82 views

A problem of four conics

I found a remarkable theorem of four conics as follows some years ago. But it has no proof; I am looking for a proof: Theorem: Take three conics. Suppose that each of them touch a fourth conic at two ...
0
votes
0answers
28 views

$m-$cycles in $S_n$ modulo an equivalence relation

Let $A$ be the set of all $m-$cycles in $S_n$. Define an equivalence relation $i$ in $A$ by $\sigma_1$ is related to $\sigma_2$ by $i$ if $\sigma_1$ is a power of $\sigma_2$ or viz., then the number ...
8
votes
1answer
206 views

Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?

Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$? Do there exist ...
0
votes
0answers
72 views

The prime number matrix sieve [on hold]

I have derived the following theorem: An odd positive integer $N=6n−1$ is a prime iff neither of two diophantine equations $6x^2+(6x−1)y=n$ $6x^2+(6x+1)y=n$ has a solution. An odd positive ...
1
vote
0answers
34 views

Non-zero homomorphism from a module to its ground ring

Let $c_1,\dots,c_k$ be some non-zero complex numbers and $M$ be the abelian subgroup generated by $c_1,\dots,c_k$ (i.e. all $\mathbb{Z}$-linear combinations of $c_1\dots,c_k$). Suppose further that $\...
9
votes
0answers
161 views

Fundamental circuit characterization of matroid independence complexes

I have the following characterization of independence complexes of matroids, which I think is standard but I can't find a reference. Here it goes: A pure simplicial complex $\Delta$ is the ...
6
votes
1answer
231 views

The universal property of composition of morphisms

$\def\K{\mathcal K}$ Preamble. Given a locally small category $\mathcal K$, its "composition law" is a class of maps $$ c_{abc} : \K(a,b)\times\K(b,c)\to \K(a,c) $$ with the universal property of an ...
2
votes
0answers
34 views

Oscillation operator of a function

Call a function from $[0, 1]$ to itself a box function. Given any box function $f$, define its oscillation function Of as $$Of(x) = \lim _{d \to 0} \sup _{y, z \in B_d (x)} |f(y) - f(z)|$$ Then $Of(x)...
2
votes
0answers
66 views

Bounds on the L^1 norm of a discrete Fourier spectrum

I am dealing with a function $f$ of the form \begin{equation} f(t):=\sum_{k=1}^Na_ke^{\mathrm{i}\phi_k t} \end{equation} and I have a promise that \begin{equation} 0\leq f(t)\leq C\;\;\;\text{for all}...
1
vote
1answer
64 views

Relative weight lattice

Let $G$ be a reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus, $B$ be a Borel subgroup and $I_G$ is the set of simple roots. Let $P$ be a parabolic subgroup, $M$ be ...
0
votes
0answers
17 views

Optimal transport between Gaussian mixtures and their centers

I have a question about bounding the Wasserstein loss between a continuous Gaussian mixture and a discrete uniform distribution of its centers. In particular, let $P=\frac 1 k \sum_{i=1}^k \mathcal{N}(...
3
votes
1answer
58 views

Globalizing Feigin--Frenkel duality

Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{g}^L$ be its Langlands dual. Feigin--Frenkel duality says $$ W^k(\mathfrak{g})=W^{k_L}(\mathfrak{g}^L) $$ if $r'(k+h^{'})(k_L+h'_L)=1$, where ...
1
vote
1answer
61 views

Does $f$ have the same minimiser as $\|\nabla f \|$ for $f$ strictly convex?

This question is migrated from MathStackExchange where it seemed to be too hard. I wonder does anyone here have any ideas? Suppose $f: K \to \mathbb R$ is $\mathcal C^2$ and strictly convex on some ...
1
vote
1answer
101 views

On $\det[x+(\frac{i\pm j}p)]_{1\le i,j\le(p-1)/2}$ for primes $p\equiv 3\pmod 4$

I have made the followng conjecture on the basis of my computation. Conjecture. For any prime $p\equiv3\pmod4$ with $p>3$, we have $$\det\left[x+\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}...
1
vote
0answers
52 views

Final time maps of IVP's approximating functions $X\subseteq\mathbb{R}^n\to\mathbb{R}^n$

I originally posted this question on the Mathematics StackExchange and got told to consider putting it on here, on MathOverflow. I will word the question a bit differently: Let $X$ be a compact $k$-...
3
votes
1answer
196 views

does recursive (decidable) languages closed under division (Quotient) with any language?

I need to prove or disprove that R languages are closed under divison. I have managed to prove thet CFL are't closed under division. I read in wikipedia that RE languages are closed, but I didn't find ...
0
votes
0answers
48 views

Give an example of a paracompact space whose has finite cohomological dimension

Borel-Atiyah-Quillen-Hsiang localization theorem is stated for compact Lie group actions on compact spaces or paracompact spaces of finite cohomological dimension. In fact, Satya Deo generalized the ...
0
votes
1answer
77 views

Underdetermined system of linear PDEs

Let $a,b$ two smooth functions from the open square square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$, and let $a^{i},b^{i}$ denote the $i$-th component functions. I need to study the PDE ...
0
votes
0answers
35 views

Optimizing a complex functional with respect to the Lexicographic Ordering?

I'm wondering if the following argument is correct: Consider optimizing a complex functional $S[x(t)]$. Since $S$ is complex, it only has an optimum with respect to the lexicographic order of the ...
1
vote
0answers
51 views

Is the set of fixed points of hyperbolic elements from $\mathrm{PSL}_{2}(A)$ a group?

Given a subring $A$ of $\mathbb{R}$, we can consider the set $\mathrm{PSL}_{2}(A)$ of elements in $\mathrm{PSL}_{2}(\mathbb{R})$ with entries in $A$ and the determinant of associated matrix is equal ...
4
votes
1answer
148 views

Half-dimensional torus fibration vs Lagrangian torus fibration

Assume we have a closed symplectic manifold $M$ which is the total space of a smooth fibration by half-dimensional tori. Can we infer that $M$ is the total space of a smooth fibration by Lagrangian ...
4
votes
1answer
105 views

Can smoothness of curves into a convenient locally convex vector space be tested with just a dense subspace of the dual?

Let $E$ be a (Hausdorff) locally convex vector space (from now on just "lcs" for short). We say that $E$ is convenient (also called locally complete, Mackey-complete or $c^\infty$-complete) if, given ...
1
vote
0answers
25 views

Decomposition into irreducible of a representation of the wreath product $S_d \wr S_m$ (2)

This is a question following Decomposition into irreducible of a representation of the wreath product $S_d\wr S_n$ I call: $$ R_m= \bigl( F^{\widetilde{\otimes n-m}} \boxtimes S^{\widetilde{\otimes m}...
7
votes
1answer
238 views

Is there a conceptual reason why the notion of “quasicoherent sheaf” is independent of the choice of topology?

Let $X$ be a scheme and $\mathcal S$ a site which is a full subcategory of the category $Aff/X$ of affine schemes with a map to $X$. If I understand correctly, the category $QCoh^\mathcal S(X)$ of $\...
5
votes
1answer
94 views

Kato's Euler System for Isogenous Elliptic Curves

Let $E,E^\prime$ be elliptic curves over $\mathbb{Q}$ and also suppose they are $p$-isogenous. How are the Euler systems corresponding to the two isogenous elliptic curves related, if at all?
7
votes
0answers
75 views

What does the classifying space of a topological monoid classify?

The classifying space $BG$ of a topological group $G$ classifies principal $G$ bundles. I have come to appreciate this. I hope the following question is appropriate for MathOverflow: What does the ...
-1
votes
0answers
37 views

How to understand the sign periodicity of any conditionally convergent series? [on hold]

Given any conditionally convergent series $\sum_{n\geq1} a_n$. I wonder if there is a "standard way/method" to "investigate/estimate" the sign periodicity of $a_n$ by some explicit functions $f(x)$ i....
-4
votes
0answers
43 views

what are the math books worth to read published in 2018? [on hold]

Holidays are approaching fast and books are always great gifts. By math books, I mean undergraduate textbooks, large audience math books, specific field books. The only request is that they have ...
3
votes
2answers
305 views

Distance between primes that are quadratic residues modulo an other prime

Question: Is there an infinite sequence of primes $\{q_i\}_{i=1}^{\infty}$ that is not too sparse ( $q_n =O(poly(n))$ for a fixed polynomial) for which it is true that for every $k$ there is an $N(k)$ ...
2
votes
0answers
34 views

A question related to (random) matrix factorization

Let $\mathbf{B}$ be an $m\times r$ binary matrix taking values in $\{0,1\}$, and assume that $m\geq r$. I am wondering what can be said about the recovery of $\mathbf{B}$ from $\mathbf{B}\mathbf{B}^T$....

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